Problem 4 - Algebra and Trigonometry

Question

(a) The average rate of change of a function \(f\) between \(x=a\) and \(x=b\) is the slope of the _____ line between \((a, f(a))\) and \((b, f(b))\). (b) The average rate of change of the linear function \(f(x)=3 x+5\) between any two points is _____.

Step-by-Step Solution

Answer

(a) secant; (b) 3.

Step 1: Understanding Average Rate of Change

The average rate of change of a function between two points is essentially the slope of the line that connects those points. This line is known as the secant line. So, for part (a), the average rate of change between \(x=a\) and \(x=b\) is the slope of the secant line between \((a, f(a))\) and \((b, f(b))\).

Step 2: Calculating Average Rate of Change for Linear Functions

For a linear function, the average rate of change between any two points is constant and equal to the slope of the line defined by the function. In the case of the function \(f(x)=3x+5\), its slope is 3. Therefore, the average rate of change between any two points for this function is 3.

Key Concepts

Secant Line

A secant line is a straight line that connects two points on the graph of a function. When we talk about the average rate of change of a function between two points, we are essentially discussing the slope of the secant line that runs between those points. Imagine plotting two points on a curve, for example, at coordinates

• \((a, f(a))\)
• \((b, f(b))\)

To find the secant line, draw a straight line connecting these two points. This line gives us a simple, visual representation of how much the function is changing, on average, between those points.
By calculating the slope of the secant line, we can find the average rate of change. This is especially useful when dealing with functions that vary erratically, as the secant line simplifies the analysis by focusing on just two points.

Linear Function

Linear functions are the simplest type of functions, typically represented in the form \(f(x) = mx + c\), where \(m\) and \(c\) are constants. The graph of a linear function is always a straight line, which makes it easy to understand and analyze. Because they are straight lines, the average rate of change across the entire function is constant and is simply the slope \(m\). This constancy means that the secant line isn't just connecting two points—it's equivalent to the entire graph!
For instance, consider the linear function \(f(x) = 3x + 5\). The slope \(m = 3\) is the same everywhere on this line, so wherever you choose two points, the average rate of change (or the slope of the secant line) remains the constant value of 3. Linear functions make it particularly straightforward to calculate average rates of change due to this inherent property.

Slope Calculation

The slope of a line is a measure of how steep the line is. It's calculated using the formula:\[slope = \frac{f(b) - f(a)}{b - a}\]This formula gives us the rate at which \(y\) changes for a given change in \(x\). It essentially tells us how much the function value \(f(x)\) increases or decreases as \(x\) moves from \(a\) to \(b\).
For linear functions, slope calculation is particularly simple because the result is the same regardless of the points chosen. Using the linear function example \(f(x) = 3x + 5\), substitute any two points into the formula. For example, let's calculate the slope between points where \(x = 1\) and \(x = 2\):

• \(f(1) = 3 \times 1 + 5 = 8\)
• \(f(2) = 3 \times 2 + 5 = 11\)

So, the slope:\[\frac{11 - 8}{2 - 1} = 3\]This confirms the consistency of the slope across any two points of a linear function.